Is it possible to show that: \begin{align} \frac{\Gamma\left(\frac{1}{78}\right) \Gamma\left(\frac{29}{78}\right) \Gamma\left(\frac{35}{78}\right) \Gamma\left(\frac{53}{78}\right) \Gamma\left(\frac{55}{78}\right) \Gamma\left(\frac{61}{78}\right)}{\Gamma\left(\frac{2}{78}\right) \Gamma\left(\frac{28}{78}\right) \Gamma\left(\frac{32}{78}\right) \Gamma\left(\frac{44}{78}\right) \Gamma\left(\frac{58}{78}\right) \Gamma\left(\frac{70}{78}\right) } = \sqrt{3} \end{align}
There are other known ratios of Gamma functions, but as always there seems to be a product rule or trick to evaluate the ratios.
One method to find such formulas is the duplication formula $$ \Gamma(z) \Gamma(z+1/2) = 2^{1-2z} \sqrt{\pi} \Gamma(2z), $$ and apply it to nominator and denominator, e.g., to obtain $$ {\Gamma(1/8) \Gamma(5/8) \Gamma(6/8) \over \Gamma(2/8) \Gamma(3/8) \Gamma(7/8)} = \sqrt{2}. $$ There seem to be more advanced ideas in Deligne, P. Valeurs de fonctions L et périodes d'intégrales, Amer. Math. Soc., Providence, R.I., 1979.